Signal Processing for Remote Sensing - Chapter 5 pdf

5 Multi-Dimensional Seismic Data Decomposition by Higher Order SVD and Unimodal ICA Nicolas Le Bihan, Valeriu Vrabie, and Je ´ ro ˆ me I. Mars CONTENTS 5.1 Introduction 74 5.2 Matrix Data Sets 74 5.2.1 Acquisition 75 5.2.2 Matrix Model 75 5.3 Matrix Processing 76 5.3.1 SVD 76 5.3.1.1 Definition 76 5.3.1.2 Subspace Method 76 5.3.2 SVD and ICA 77 5.3.2.1 Motivation 77 5.3.2.2 Independent Component Analysis 77 5.3.2.3 Subspace Method Using SVD–ICA 79 5.3.3 Application 80 5.4 Multi-Way Array Data Sets 83 5.4.1 Multi-Way Acquisition 84 5.4.2 Multi-Way Model 84 5.5 Multi-Way Array Processing 85 5.5.1 HOSVD 85 5.5.1.1 HOSVD Definition 85 5.5.1.2 Computation of the HOSVD 86 5.5.1.3 The (r c , r x , r t )-rank 87 5.5.1.4 Three-Mode Subspace Method 88 5.5.2 HOSVD and Unimodal ICA 88 5.5.2.1 HOSVD and ICA 89 5.5.2.2 Subspace Method Using HOSVD–Unimodal ICA 89 5.5.3 Application to Simulated Data 90 5.5.4 Application to Real Data 95 5.6 Conclusions 98 References 98 ß 2007 by Taylor & Francis Group, LLC. 5.1 Introduction This chapter describes multi-dimensional seismic data processing using the higher order singular value decomposition (HOSVD) and partial (unimodal) independent component analysis (ICA). These techniques are used for wavefield separation and enhancement of the signal-to-noise ratio (SNR) in the data set. The use of multi-linear methods such as the HOSVD is motivated by the natural modeling of a multi-dimensional data set using multi- way arrays. In particular, we present a multi-way model for signals recorded on arrays of vector-sensors acquiring seismic vibrations in different directions of the 3D space. Such acquisition schemes allow the recording of the polarization of waves and the proposed multi-way model ensures the effective use of polarization information in the processing. This leads to a substantial increase in the performances of the separation algorithms. Befo re in troducing the mu lti-way mo del and process ing, we first describe the classic al subsp ace method based on the SVD and ICA techn iques for 2D (mat rix) seismic data sets. Using a matrix model for these data sets, the SV D-bas ed subsp ace me thod is pres ented and it is shown how an extra ICA step in the pr ocessin g allows bette r wave field separ- ation. Then, conside ring sign als recorded on vector- sensor arrays , the multi-wa y mode l is define d and discusse d. The HOSVD is pre sented and som e proper ties det ailed. Bas ed on this multi- linear decomp osition, we propose a subspace method that allows separ ation of polarize d wave s unde r orthogo nality co nstrai nts. We then introduce an ICA step in the pro cess that is perform ed here uni quely on the temp oral mode of the data set, leading to the so-call ed HOSV D–unim odal ICA subsp ace algorit hm. Resul ts on sim ulated and real polarize d data sets sho w the ability of this algorit hm to surpas s a matr ix-based algorithm and subspace method usin g only the HOSVD. Sectio n 5.2 pre sents matr ix da ta sets and their associa ted mod el. In Section 5.3, the well- known SVD is detailed, as well as the matrix-base d subspace method. The n, we pr esent the ICA co ncept and its contrib ution to subspace formulat ion in Section 5.3.2. App lica- tions of SVD–ICA to seismic wave field separatio n are discussed by way of illu stration s. Sectio n 5.4 exp oses how sign al mixtur es recorded on vecto r-sens or array s can be desc ribed by a mult i-way mod el. Then, in Se ction 5.5, we introdu ce the HO SVD and the associa ted subspace me thod for multi-wa y data proces sing. As in the matrix data set case, an extra ICA step is proposed leading to a HOSVD–unimodal ICA subspace method in Section 5.5.2. Final ly, in Sectio n 5.5.3 and Section 5.5 .4, we illustrat e the propose d algorithm on simulated and real multi-way polarized data sets. These exampl es empha- size the potential of using both HOSVD and ICA in multi-way data set processing. 5.2 Matrix Data Sets In this section, we show how the signals recorded on scalar-sensor arrays can be modeled as a matrix data set having two modes or diversities: time and distance. Such a model allows the use of subspace-based processing using a SVD of the matrix data set. Also, an additional ICA step can be added to the processing to relax the unjustified orthogonality constraint for the propagation vectors by imposing a stronger constraint of (fourth-order) independence of the estimated waves. Illustrations of these matrix algebra techniques are presented on a simulated data set. Application to a real ocean bottom seismic (OBS) data set can be found in Refs. [1,2]. ß 2007 by Taylor & Francis Group, LLC. 5.2.1 Acquisition In geophysics, the most commonly used method to describe the structure of the earth is seismic reflection. This method provides images of the underground in 2D or 3D, depending on the geometry of the network of sensors used. Classical recorded data sets are usually gathered into a matrix having a time diversity describing the time or depth propagation through the medium at each sensor and a distance diversity related to the aperture of the array. Several methods exist to gather data sets and the most popular are common shotpoint gather, common receiver gather,orcommon midpoint gather [3]. Seismic processing consists in a series of elementary processing procedures used to transform field data, usually recorded in common shotpoint gather into a 2D or 3D common midpoint stacked 2D signals. Before stacking and interpretation, part of the processing is used to suppress unwanted coherent signals like multiple waves, ground-roll (surface waves), refracted waves, and also to cancel noise. To achieve this goal, several filters are classically appli ed on seismic data sets. The SVD is a popular method to separate an initial data set into signal and noise subspaces. In some applications [4,5] when wavef ield alignment is performed, the SVD me thod allows separation of the aligned wave from the other wavefields. 5.2.2 Matrix Model Consider a uniform linear array composed of N x omni-directional sensors recording the contributions of P waves, with P < N x . Such a record can be written mathematically using a convolutive model for seismic signals first suggested by Robinson [6]. Using the superposition principle, the discrete-time signal x k (m)(m is the time index) recorded on sensor k is a linear combination of the P waves received on the array together with an additive noise n k (m): x k (m) ¼ X P i¼1 a ki s i (m À m ki ) þ n k (m)(5:1) where s i (m) is the ith source waveform that has been propagated through the transfer function supposed here to consist in a delay m ki and a factor attenuation a ki . The noises on each sensor n k (m) are supposed centered, Gaussian, spatially white, and independent of the sources. In the sequel, the use of the SVD to separate waves is only of significant interest if the subspace occupied by the part of interest contained in the mixture is of low rank. Ideally, the SVD performs well when the rank is 1. Thus, to ensure good results of the process, a preprocessing is applied on the data set. This consists of alignment (delay correction) of a chosen high amplitude wave. Denoting the aligned wave by s 1 (m), the model becomes after alignment: y k (m) ¼ a k1 s 1 (m) þ X P i¼2 a ki s i (m À m 0 ki ) þ n 0 k (m)(5:2) where y k (m) ¼ x k (m þ m k1 ), m ki 0 ¼ m ki À m k1 and n 0 k (m) ¼ n k (m þ m k1 ). In the following we assume that the wave s 1 (m) is independent from the others and therefore independent from s i (m À m ki 0 ). ß 2007 by Taylor & Francis Group, LLC. C onsider ing t he sim plifi ed model of th e re ceiv ed signa ls ( Equa tion 5 .2 ) a nd supposin g N t time samples available, we define the matrix model of the recorded data set Y 2 R N x  N t as Y ¼ {y km ¼ y k (m)j 1 k N x ,1 m N t }(5:3) That is, the data matrix Y has rows that are the N x signals y k (m) given in Equation 5.2. Such a model allows the use of matrix decomposition, and especially the SVD, for its processing. 5.3 Matrix Pr ocessing We now present the definition of the SVD of such a data matrix that will be of use for its decomposition into orthogonal subspaces and in the associated wave separation technique. 5.3.1 SVD As the SVD is a widely used matrix algebra technique, we only recall here theoretical remarks and redirect readers interested in computational issues to the Golub and Van Loan book [7]. 5.3.1.1 Definition Any matrix Y 2 R N x  N t can be decomposed into the product of three matrices as follows: Y ¼ UDV T (5:4) where U is a N x  N x matrix, D is an N x  N t pseudo-diagonal matrix with singular values {l 1 , l 2 , ,l N } on its diagonal, satisfying l 1 ! l 2 ! ! l N ! 0, (with N ¼ min(N x , N t )), and V is an N t  N t matrix. The columns of U (respectively of V) are called the left (respectively right) singular vectors, u j (respectively v j ), and form orthonormal bases. Thus U and V are orthogonal matrices. The rank r (with r N) of the matrix Y is given by the number of nonvanishing singular values. Such a decomposition can also be rewritten as Y ¼ X r j¼1 l j u j v T j (5:5) where u j (respectively v j ) are the columns of U (respective ly V). This notation shows that the SVD allows any matrix to be expressed as a sum of r rank-1 matrices 1 . 5.3.1.2 Subspace Method The SVD has been widely used in signal processing [8] because it gives the best rank approximation (in the least squares sense) of a given matrix [9]. This property allows denoising if the signal subspace is of relatively low rank. So, the subspace method consists of decomposing the data set into two orthogonal subspaces with the first one built from the p singular vectors related to the p highest singular values being the best rank approximation of the original data. This can be written as follows, using the SVD notation used in Equation 5.5, for a data matrix Y with rank r: 1 Any matrix made up of the product of a column vector by a row vector is a matrix whose rank is equal to 1 [7]. ß 2007 by Taylor & Francis Group, LLC. Y ¼ Y Signal þ Y Noise ¼ X p j ¼ 1 l j u j v T j þ X r j ¼ pþ 1 l j u j v T j (5: 6) Orthogo nality betwee n the subsp aces spanned by the two sets of singular vecto rs is ensu red by the fact that left and rig ht singular v ectors form orthon ormal bases. From a practica l point of view, the val ue of p is chosen by finding an abrup t chan ge of slope in the cu rve of relativ e sin gular values (relative meani ng perce ntile repres entation of) contain ed in the matrix D defined in Eq uation 5.4. For some case s wh ere no ‘‘visible ’’ change of slope can be found, the value of p can be fixed at 1 for a perfect alignment of waves, or at 2 for an imperfect alignment or for dispersive waves [10]. 5.3.2 SVD and ICA The motivation to relax the unjustified orthogonality constraint for the propagation vectors is now presented. ICA is the method used to achieve this by imposing a fourth- order independence on the estimated waves. This provides a new subspace method based on SVD–ICA. 5.3.2.1 Motivation The SVD of the data matrix Y in Equation 5.4 provides two orthogonal matrices composed by the left u j (respectively right v j ) singular vectors. Note here that v j are called estimated waves because they give the time dependence of received signals by the array sensor and u j propagation vectors because they give the amplitude of v j 0 s on sensors [2]. As SVD provides orthogonal matrices, these vectors are also orthogonal. Orthogonality of the v j 0 s means that the estimated waves are decorrelated (second-order independence). Actually, this supports the usual cases in geophysical situations, in which recorded waves are supposed decorrelated. However, there is no phys ical reason to consider the ortho- gonality of propagation vectors u j . Why should we have different recorded waves with orthogonal propagation vectors? Furthermore, imposing the orthogonality of u j 0 s, the estimated waves v j are forced to be a mixture of recorded waves [1]. One way to relax this limitation is to impose a stronger criterion for the estimated waves, that is, to be fourth-order statistically independent, and consequently to drop the unjustified orthogonality constraint for the propagation vectors. This step is motivated by cases encountered in geophysical situations, where the recorded signals can be approxi- mated as an instantaneous linear mixture of unknown waves supposed to be mutually independent [11]. This can be done using ICA. 5.3.2.2 Independent Component Analysis ICA is a blind decomposition of a multi-channel data set composed of an unknown linear mixture of unknown source signals, based on the assumption that these signals are mutually statistically independent. It is used in blind source separation (BSS) to re- cover independent sources (modeled as vectors) from a set of recordings containing linear combinations of these sources [12–15]. The statistical independence of sources means that the cross-cumulants of any order vanish. Generally, the third-order cumu- lants are discarded because they are generally close to zero. Therefore, here we will use fourth-order statistics, which have been found to be sufficient for instantaneous mixtures [12,13]. ß 2007 by Taylor & Francis Group, LLC. ICA is usuall y res olved by a two- step algor ithm: pre whiteni ng follow ed by high- order step. The first one co nsists in extra cting decorrel ated waves from the initia l data set. The step is carri ed out direc tly by an SV D as the v j 0 s are orthogon al. The second step consists in finding a rotatio n matrix B , whic h leads to fou rth-order inde penden ce of the estimate d waves . We suppos e here that the nonaligne d waves in the data set Y are containe d in a subspace of dim ension R À 1, smaller than the ran k r of Y. Assumi ng this , on ly the first R estimate d waves [v 1 , , v R ] notation ¼ V R 2 R N t  R are take n into acco unt [2]. As the recorde d waves are suppo sed mu tually indepen dent, this sec ond step can be writte n as V R B ¼ ~ VV R ¼ [ ~ vv 1 , , ~ vv R ] 2 R N t  R (5 :7) with B 2 R R  R the rotatio n (unitar y) matr ix having the proper ty BB T ¼ B T B ¼ I . The new estimate d waves ~ vv j are no w inde pendent at the fou rth order. The re are differen t me thods of findi ng the rotatio n matrix: joint appr oximate diago na- lizat ion of eige nmatrices (JADE ) [12] , maxim al diagon ality (MD) [13], sim ultane ous third- orde r tensor diago nalizatio n (STOTD ) [14], fast and robu st fixed -point algor ithms for inde penden t co mponent analys is (FastI CA) [15], and so on. To compare som e ci ted ICA algor ithms, Figure 5.1 sho ws the rela tive error (see Equa tion 5.12) of the estimate d sign al subsp ace versu s the SN R (see Eq uation 5.11) for the data set pre sented in Section 5.3. 3. For SN Rs gre ater than À7.5 dB, Fa stICA usin g a ‘‘ tan h’ ’ no nlinearity with the parameter equal to 1 in the fixed-poi nt algor ithm pro vides the smallest relative er ror, but with som e erroneo us points at differen t SNR. Note that the ‘‘tan h’’ nonli nearity is the one which gives the smal lest error for this data set, co mpared with ‘‘pow3’ ’, ‘‘g auss’’ with the parameter equal to 1, or ‘‘s kew’’ nonli nearities. MD and JA DE algorithm s are appr oxi- matel y equi valent accordin g to the relati ve error. For SNRs smaller than À 7.5 dB, MD pro vides the smallest relative error. Consequently, the MD algorithm was employed in the following. Now, consi dering the SV D dec ompositi on in Equa tion 5.5 and the ICA step in Equa tion 5.7, the subspace described by the first R estimated waves can be rewritten as 6 5 4 3 Error (%) of the estimated signal subspace 2 1 0 –20 –15 –10 –5 0 SNR (dB) of the dataset 51015 JADE FastlCA–tan h MD FIGURE 5.1 ICA algorithms—comparison. ß 2007 by Taylor & Francis Group, LLC. X R j ¼ 1 l j u j v T j ¼ U R D R V R ÀÁ T ¼ U R D R B ~ VV R ÀÁ T ¼ X R j¼ 1 b j ~ uu j ~ vv T j ¼ X R i¼ 1 b i ~ uu i ~ vv T i (5: 8) where U R ¼ [ u 1 , , u R ] is made up of the first R vectors of the matrix U and D R ¼ diag ( l 1 ,. , l R ) is the R  R trun cated version of D contain ing the gre atest values of l j . The second equality is obta ined usin g Equa tion 5.7. For the third equality, the ~ uu j are the new propa gation vectors obtained as the norm alized 2 vec tors (col umns) of the matrix U R D R B and b j are the ‘‘modi fied sing ular values’ ’ obt ained as the ‘ 2 -nor m of the co lumns of the matrix U R D R B. The elemen ts b j are usual ly not ord ered. For this reason, a per mutation betwee n the vectors ~ uu j as wel l as bet ween the vectors ~ vv j is per formed to order the modifi ed singular values. Denotin g with s( Á ) this permu tation and with i ¼s( j), the last equali ty of Equa tion 5.8 is obtaine d. In this decom position, wh ich is similar to that given by Equa tion 5.5, a stronger criterion for the new estim ated waves ~ vv i has been im posed, that is, to be inde pendent at the fou rth ord er, and, at the same time, the condi tion of orthogon ality for the new propa gation vecto rs ~ uu i has been rela xed. In practica l situation s, the value of R become s a parameter . Usual ly, it is chosen to compl etely describ e the align ed wave by the first R estim ated waves given by the SVD. 5.3.2 .3 Su bspace Method Using SVD–IC A After the ICA and the per mutatio n steps, the sign al subspace is given by ~ YY Signal ¼ X ~ pp i ¼ 1 b i ~ uu i ~ vv T i (5: 9) where ~ pp is the number of the new estimated waves necessary to describe the aligned wave. The noise subsp ace ~ YY Noise is obt ained by subt raction of the sign al subspace ~ YY Signal from the original da ta set Y : ~ YY Noise ¼ Y À ~ YY Signal (5: 10) From a practica l point of view , the value of ~ pp is chosen by findi ng an ab rupt change of slope in the curve of relati ve mo dified sin gular values. For case s with low SN R, no ‘‘visible ’’ change of slope can be found and the value of ~ pp can be fixed at 1 for a perfect align ment of waves, or at 2 for an imper fect align ment or for dispersive waves. Note here that for ver y smal l SN R of the initial data set, (for ex ample, smaller than À 6.2 dB for the data set pre sented in Section 5.3.3, the align ed wave can be describ ed by a less energe tic estim ated wave than by the first one (related to the highes t sin gular value) . For these extrem e cases , a search mu st be done after the ICA and the per mutatio n steps to iden tify the indexe s for which the corresp onding estim ated waves ~ vv i give the aligned wave. So the signal subsp ace ~ YY Signal in Equati on 5.9 must be rede fined by choosing the inde x values fo und in the search . For exampl e, applying the MD algor ithm to the data set pres ented in Section 5.3.3 for which the SNR was mod ified to À9 dB, the align ed wave is desc ribed by the third estim ated wave ~ vv 3 . Note also that using SVD without ICA in the same conditions, the aligned wave is described by the eighth estimated wave v 8 . 2 Vectors are normalized by their ‘ 2 -norm. ß 2007 by Taylor & Francis Group, LLC. 5.3. 3 Applic ation An applicati on to a sim ulated data set is pre sented in this secti on to illus trate the beh avior of the SVD–I CA versu s the SVD subspace me thod. Applic ation to a real da ta set obtaine d duri ng an acqui sition with OBS can be fou nd in Ref s. [1,2] . The preprocessed recorded signals Y on an 8-sensor array (N x ¼ 8) during N t ¼ 512 time samples are represented in Figure 5.2c. This synthetic data set was obtained by the addition of an original signal subspace S (Figure 5.2a) made up by a wavefront having infinite celerity (velocity), consequently associated with the aligned wave s 1 (m), and an original noise sub- space N (Figure 5.2b) made up by several nonaligned wavefronts. These nonaligned waves are contain ed in a subspace of dimens ion 7, sm aller than the rank of Y, whic h equals 8. The SNR ratio of the pre sented da ta set is SNR ¼À3.9 dB. The SNR definiti on used here is 3 : SNR ¼ 20 log 10 kSk kNk (5 :11) Norm alizat ion to unit varian ce of eac h trace for eac h compo nent was don e bef ore apply ing the descri bed subspace methods . This ens ures that even weak picked arrivals are well repre sented within the inp ut da ta. Afte r the comp utatio n of sign al subspaces, a deno rmalizatio n was applie d to find the origin al signal subsp ace. Firs tly, the SV D subspace me thod was teste d. The subsp ace method given by Equa tion 5.6 was emp loyed, keepi ng only one sin gular vec tor (resp ective ly one sin gular val ue). This choice was mad e by finding an abrupt chan ge of slop e after the first singular value (Fig ure 5.6) in the relati ve sing ular val ues for this da ta set. The obtain ed sign al subspace Y Signal and noise subspace Y Noise are presented in Figure 5.3a and Figure 5.3b . It is clear Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (a) Original signal subspace S Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (b) Original noise subspace N Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (c) Recorded signals Y = S ϩ N FIGURE 5.2 Synthetic data set. 3 kAk¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S I i¼1 S J j¼1 a 2 ij q is the Frobenius norm of the matrix A ¼ {a ij } 2 R I  J ß 2007 by Taylor & Francis Group, LLC. from thes e figures that the classical SVD im plies artifa cts in the two estim ated subspace s for a wavefield separ ation obje ctive. Moreove r, the estim ated waves v j shown in Figure 5.3c are an instan taneous linear mixture of the recorde d wave s. Th e s ig na l s ubsp ac e ~ YY Si gn al and noise subspace ~ YY Noise obtained using the SVD–ICA subspace method given by Equation 5.9 are presented in Figure 5.4a and Figure 5.4b. This improvement is due to the fact that using ICA we have imposed a fourth-order independence condition stronger than the decorrelation used in classical SVD. With this subspace method we have also relaxed the nonphysically justified orthogonality of the propagation vectors. The dimen sion R of the rotati on matr ix B was chos en to be eight becau se the aligned waveli ght is pro jected on all eight estimate d waves v j shown in Figure 5.3c. Afte r the ICA and the permutatio n steps, the new estimate d waves ~ vv i are pre sented in Figure 5.4c. As we can see, the first one desc ribes the aligned wave ‘‘per fectly’ ’. As no visible chan ge of slope can be found in the rela tive modified sing ular values sho wn in Figure 5.6, the value of ~ pp was fixed at 1 because we are deali ng with a per fectly aligned wave. To compa re the res ults qualitat ively, the stack rep resentat ion is usually employe d [5]. Figure 5.5 sho ws, from left to right, the stacks on the initial da ta set Y, the origin al sign al subspace S , and the estim ated sign al subsp aces obtain ed with SVD and SV D–ICA sub- space me thods, respecti vely. As the stack on the estima ted signal subspace ~ YY Sig nal is very close to the stack on the origin al sign al subspace S, we can co nclude that the SVD–I CA subspace me thod enhan ces the wave separatio n res ults. To comp are thes e methods quantitat ively, we use the relative error « of the estimate d signal subspace define d as «¼ kS À  YY Si gnal k 2 kSk 2 (5: 12) Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (a) Estimated signal subspace Y Signal Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (b) Estimated noise subspace Y Noise Index (i ) 1 2 3 4 5 6 7 8 500 100 150 200 250 300 Time (samples) 350 400 450 500 (c) Estimated waves v j FIGURE 5.3 Results obtained using the SVD subspace method. ß 2007 by Taylor & Francis Group, LLC. Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (a) Estimated signal subspace Y Signal ~ Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 500 (b) Estimated noise subspace Y Noise ~ Index (i ) 1 2 3 4 5 6 7 8 500 100 150 200 250 300 Time (samples) 350 400 450 500 (c) Estimated waves v i ~ FIGURE 5.4 Results obtained using the SVD-ICA subspace method. FIGURE 5.5 Stacks. From left to right: initial data set Y, original signal subspace S, SVD, and SVD–ICA estimated subspaces. 0 50 100 150 250 Time (samples) 350200 300 400 450 500 ß 2007 by Taylor & Francis Group, LLC. [...]... 9 estimated waves v(t)j shown in Figure 5. 19a As suggested, R becomes a parameter ß 2007 by Taylor & Francis Group, LLC Distance (m) Distance (m) 0 50 100 150 200 250 300 350 400 450 50 0 0 0 50 100 150 200 250 300 350 400 450 50 0 0 0 50 100 150 200 250 300 350 400 450 50 0 0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.4 Time (s) 0.4 Time (s) 0.3 0.4 Time (s) 0.3 0 .5 0 .5 0 .5 0.6 0.6 0.6 0.7 0.7 0.7 (a) X component... SVD–ICA 25 20 15 10 5 0 1 2 3 4 5 Index ( j ) 6 7 8 FIGURE 5. 6 Relative singular values where kÁk is the matrix Frobenius norm defined above, S is the original signal subspace  and YSignal represents either the estimated signal subspace YSignal obtained using SVD or ~ the estimated signal subspace YSignal obtained using SVD–ICA For the data set presented in Figure 5. 2, we obtain « ¼ 55 .7% for classical... component 60 50 40 1 30 20 10 0 1 15 10 5 0 1 2 2 4 5 6 10 8 15 10 12 20 14 25 16 30 18 36 (a) Using HOSVD 60 50 40 1 30 20 10 0 1 15 10 5 0 1 2 2 4 5 6 10 8 15 10 20 12 14 25 (b) Using HOSVD−unimodal ICA 16 30 18 36 FIGURE 5. 17 Relative three-mode singular values ~ As the stack on the estimated signal subspace YSignal is very close to the stack on the original signal subspace S, we conclude that the... 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 0 0 50 50 50 100 150 150 150 Time (samples) 100 Time (samples) 100 Time (samples) 200 200 200 250 250 250 (b) Original noise subspace N (a) Original signal subspace S (c) Initial dataset Y FIGURE 5. 10 Simulated data: the Z component Our aim is to recover the original signal (Figure 5. 10a and Figure 5. 11a) from the mixture, which is, in practice, the only... the slopes for the other two modes As for the bidimensional case, to compare these methods quantitatively, we use the relative error « of the estimated signal subspace defined as ß 2007 by Taylor & Francis Group, LLC Distance (sensors) 1 2 3 4 5 0 2 4 6 8 10 12 14 0 50 50 50 100 150 150 Time (samples) 100 150 Time (samples) 100 Time (samples) 200 200 200 250 250 250 ~ (a) Z component of YSignal Index... 0 (a) Original signal subspace S FIGURE 5. 11 Simulated data: the Hy component Distance (sensors) Distance (sensors) Distance (sensors) 0 2 4 6 8 10 12 14 16 18 0 0 2 4 6 8 10 12 14 16 18 0 50 50 100 150 150 Time (samples) 100 Time (samples) 200 200 250 250 (b) Hy component (a) Z component FIGURE 5. 12 Signal subspace using component-wise SVD within the input data After computation of signal subspaces,... 0 .5 0 .5 0.6 0.6 0.7 0.7 (a) Before ICA, v(t )j ~ (b) After ICA, v(t )j 1 3 5 7 0.1 0.1 ß 2007 by Taylor & Francis Group, LLC 9 0 1 3 5 7 9 0 FIGURE 5. 19 The first 9 estimated waves 60 40 20 1 2 3 20 10 01 10 10 20 30 40 50 5 0 1 20 40 60 80 100 120 140 (a) Using HOSVD 60 40 20 1 20 2 3 10 0 1 10 10 20 30 40 50 5 0 1 20 40 60 80 100 120 140 (b) Using HOSVD–unimodal ICA FIGURE 5. 20 Relative three-mode... with a unimodal ICA step for vector-sensor array signal processing 5. 6 Conclusions We have presented a subspace processing technique for multi-dimensional seismic data sets based on HOSVD and ICA It is an extension of well-known subspace separation techniques for 2D (matrix) data sets based on SVD and more recently on SVD and ICA The proposed multi-way technique can be used for the denoising and separation... three-mode data sets recorded during Nt time samples on vector-sensor arrays made up by Nx sensors each one having Nc components can be modeled as multi-way arrays Y 2 RNc  Nx  Nt: ß 2007 by Taylor & Francis Group, LLC Y ¼ fyckm ¼ yck (m)j1 c Nc , 1 k Nx , 1 m Nt g (5: 15) This multi-way model can be used for extension of subspace method separation to multicomponent data sets 5. 5 Multi- Way A rray Processing. .. 50 100 150 150 Time (samples) 100 150 Time (samples) 100 Time (samples) 200 200 200 250 250 250 (b) Hy component of YSignal 5 0 0 2 4 6 8 10 12 14 0 (a) Z component of YSignal Index ( j ) 0 0 2 4 6 8 10 12 14 16 18 16 18 Distance (sensors) (c) Estimated waves V(t )j FIGURE 5. 14 Results obtained using the HOSVD subspace method Using the HOSVD subspace method, the components of the estimated signal subspace . HOSVD 85 5 .5. 1.1 HOSVD Definition 85 5 .5. 1.2 Computation of the HOSVD 86 5. 5.1.3 The (r c , r x , r t )-rank 87 5. 5.1.4 Three-Mode Subspace Method 88 5. 5.2 HOSVD and Unimodal ICA 88 5. 5.2.1 HOSVD. 77 5. 3.2.3 Subspace Method Using SVD–ICA 79 5. 3.3 Application 80 5. 4 Multi-Way Array Data Sets 83 5. 4.1 Multi-Way Acquisition 84 5. 4.2 Multi-Way Model 84 5. 5 Multi-Way Array Processing 85 5 .5. 1. (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 50 0 (a) Original signal subspace S Distance (sensors) 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 Time (samples) 350 400 450 50 0 (b)